Reductions of semi-stable representations using the Iwahori mod $p$ Local Langlands Correspondence
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929358749302784 |
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| author | Chitrao, Anand Ghate, Eknath |
| author_facet | Chitrao, Anand Ghate, Eknath |
| contents | We determine the mod $p$ reductions of all two-dimensional semi-stable representations $V_{k,\mathcal{L}}$ of the Galois group of $\mathbb{Q}_p$ of weights $3 \leq k \leq p+1$ and $\mathcal{L}$-invariants $\mathcal{L}$ for primes $p \geq 5$. In particular, we describe the constants appearing in the unramified characters completely. The proof involves computing the reduction of Breuil's $\mathrm{GL}_2(\mathbb{Q}_p)$-Banach space $\tilde{B}(k,\mathcal{L})$, by studying certain logarithmic functions using background material developed by Colmez, and then applying an Iwahori theoretic version of the mod $p$ Local Langlands Correspondence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_03740 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Reductions of semi-stable representations using the Iwahori mod $p$ Local Langlands Correspondence Chitrao, Anand Ghate, Eknath Number Theory 11F80 We determine the mod $p$ reductions of all two-dimensional semi-stable representations $V_{k,\mathcal{L}}$ of the Galois group of $\mathbb{Q}_p$ of weights $3 \leq k \leq p+1$ and $\mathcal{L}$-invariants $\mathcal{L}$ for primes $p \geq 5$. In particular, we describe the constants appearing in the unramified characters completely. The proof involves computing the reduction of Breuil's $\mathrm{GL}_2(\mathbb{Q}_p)$-Banach space $\tilde{B}(k,\mathcal{L})$, by studying certain logarithmic functions using background material developed by Colmez, and then applying an Iwahori theoretic version of the mod $p$ Local Langlands Correspondence. |
| title | Reductions of semi-stable representations using the Iwahori mod $p$ Local Langlands Correspondence |
| topic | Number Theory 11F80 |
| url | https://arxiv.org/abs/2311.03740 |